Given the moduli space of apparently consistent models below a given energy cut-off \(\Lambda \) , the Swampland Program [1]}, [2]}, [3]}, [4]} aims at pointing out decisive attributes of quantum field theories coupled to gravity, discerning those that admit an ultraviolet completion to quantum gravity from the others. The former ones are generally said to belong to the quantum gravity landscape, while the latter are arranged into the quantum gravity swampland. Speculative statements regarding the classification of low energy effective theories into the ones contained in the swampland and those falling into the landscape are commonly named swampland conjectures. In most occurrences of the above-mentioned discussions, the specific quantum gravity picture one has in mind is the one emerging from superstring theory. Nevertheless, evidence backing up swampland conjectures is often grounded in more general considerations on quantum field theory in curved space-time and on the expected behaviour of the quantised gravitational field. Therefore, we will refer to the ultraviolet complete theory from which landscape models are expected to descend with the vague term quantum gravity whenever possible.
The space of seemingly consistent theories below a given energy is commonly described as a geometric object, charted by a set of generalised moduli. The Distance Conjecture, which first appeared in [5]}, states that large distances in moduli space should be accompanied by infinite towers of asymptotically massless new states. Drawing inspiration from the graviton renormalization group flow in string theory \(\sigma \) -models, the distance conjecture was recently specified and rephrased [6]}, [7]}, [8]}, [9]}, [10]} in terms of geometric flow equations, such as Hamilton's Ricci flow [11]}, [12]}, [13]}
\({\left\lbrace \begin{array}{ll}\mathrm {d}_{\lambda }g_{\mu \nu }\left(\lambda \right)=-2R_{\mu \nu }\left(\lambda \right)\ ,\\g_{\mu \nu }\left(0\right)=\bar{g}_{\mu \nu }\ ,\end{array}\right.}\) 
